Eigenvalues and dynamics

An eigenvector is nonzero and returns parallel to itself after a linear map acts. Its eigenvalue gives the signed scale factor. A positive value preserves orientation, a negative value reverses it, and zero means that the direction disappears. The zero vector is excluded because it would satisfy the equation for every number and therefore identify no special direction. Eigenvectors may be multiplied by any nonzero scalar without changing their eigenvalue.

To find candidate eigenvalues of a square matrix, ask when $A-\lambda I$ has a nontrivial nullspace. Its determinant must vanish. Solving that characteristic equation supplies eigenvalues; solving the corresponding homogeneous systems supplies eigenvectors. A real matrix can have complex eigenvalues, as a planar rotation demonstrates. Working over complex numbers broadens the available modes without turning the underlying real rotation into growth.

If a matrix has a full basis of eigenvectors, express the initial state in that basis. Each update multiplies one mode coefficient by its eigenvalue. After $k$ updates, the factor is the eigenvalue raised to the $k$th power. Modes with magnitude less than one decay, those with magnitude greater than one grow, and negative values produce alternating signs. This conclusion concerns discrete updates, such as an iterative computation.

Continuous evolution follows a different rule. For the differential equation $\dot{x}=Ax$, an eigenmode evolves with an exponential $e^{\lambda t}$, so decay is controlled by the real part of the eigenvalue rather than its magnitude. Confusing these criteria would classify stable numerical updates incorrectly. A negative continuous-time eigenvalue is damping; a discrete multiplier less than negative one produces increasing oscillations.

Repeated eigenvalues do not guarantee enough independent eigenvectors. The matrix with rows $(1,1)$ and $(0,1)$ has only one independent eigenvector, and its powers contain a term that grows with the step count despite both eigenvalues equaling one. Strictly inside the unit disk, every eigenvalue still guarantees eventual decay for a finite-dimensional discrete system, but nonorthogonal modes can create substantial temporary amplification.

Real symmetric matrices avoid the missing-basis problem: they admit an orthonormal eigenbasis and real eigenvalues. Their complex counterparts are Hermitian matrices. This structural fact later makes observable operators especially useful in finite-dimensional quantum mechanics, but eigenvalues alone do not provide the measurement probabilities. Likewise, a machine-learning optimizer's local Hessian may suggest mode-dependent behavior without proving global convergence of a nonlinear training process.